Bardeen, like many of his contemporaries, followed what would now be considered a meandering, unconventional trajectory into physics, starting out as an undergrad engineer at Wisconsin, working as a geophysicist, enrolling as a math grad student at Princeton, and eventually doing a doctoral thesis with Wigner worrying about electron-electron interactions in metals (resulting in these two papers about how much energy it takes to remove an electron from a metal, and how that can be strongly affected by the very last layer of atoms at the surface - in the 1980s this would be called "surface science" and now it would be called "nanoscience").
Bardeen was a quiet, brilliant person. After WWII (during which he worked for the Navy), he went to Bell Labs, where he worked with Walter Brattain to invent the point contact transistor (and much more disagreeably with William Shockley), explaining the critical importance of "surface states" (special levels for the electrons in a semiconductor that exist at the surface, where the periodic potential of the lattice is terminated). Shockley is viewed in hindsight as famously unpleasant as a co-worker/boss - Bardeen left Bell Labs in large part because of this and ended up at Illinois, where seven years later he worked with Bob Schrieffer and Leon Cooper to produce the brilliant BCS theory of superconductivity, earning his second Nobel. (Shockley's borderline abusive management style is also responsible for the creation of modern Silicon Valley, but that's another story.)
During and after this period, Bardeen helped build the physics department of UIUC into a condensed matter physics powerhouse, a position it continues to hold. He was very interested in the theory of charge density waves (special states where the electrons in a solid spontaneously take on a spatially periodic density), though according to Lillian Hoddeson's excellent book (see here, too) he had lost the intellectual flexibility of his youth by this time.
Bardeen contributed greatly to our understanding and advancement of two whole classes of technologies that have reshaped the world (transistors and superconductors). He was not a flamboyant personality like Feynman (after all, he was from the Midwest :-) ), and he was not a self-promoter (like Feynman), but he absolutely deserves greater notoriety and appreciation from the general public.